# [cig-commits] r8148 - doc/CitcomS/manual

tan2 at geodynamics.org tan2 at geodynamics.org
Thu Oct 18 13:51:13 PDT 2007

Author: tan2
Date: 2007-10-18 13:51:12 -0700 (Thu, 18 Oct 2007)
New Revision: 8148

Modified:
doc/CitcomS/manual/citcoms.lyx
Log:
Added latent heat to the equations

Modified: doc/CitcomS/manual/citcoms.lyx
===================================================================
--- doc/CitcomS/manual/citcoms.lyx	2007-10-18 19:34:23 UTC (rev 8147)
+++ doc/CitcomS/manual/citcoms.lyx	2007-10-18 20:51:12 UTC (rev 8148)
@@ -843,7 +843,7 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-\rho c_{P}\left(T_{,t}+u_{i}T_{,i}\right)=\rho c_{P}\kappa T_{,ii}+latentheat+\rho\alpha gu_{r}\left(T+T_{0}\right)+\Phi+\rho H\label{eq:conservation of energy}$$
+\rho c_{P}\left(T_{,t}+u_{i}T_{,i}\right)=\rho c_{P}\kappa T_{,ii}+\rho\alpha gu_{r}\left(T+T_{0}\right)+\Phi+\rho Q_{L}+\rho H\label{eq:conservation of energy}

\end_inset

@@ -914,7 +914,11 @@
\begin_inset Formula $\Phi$
\end_inset

- is the viscous dissipation (TODO: latent heat unfinished) and
+ is the viscous dissipation,
+\begin_inset Formula $Q_{L}$
+\end_inset
+
+ is the latent heat, and
\emph on
H
\emph default
@@ -1070,7 +1074,7 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-x_{i}=R_{0}x_{i}^{'}\label{eq:x dim}$$
+\rho=\rho_{0}\rho^{'}\label{eq:rho dim}

\end_inset

@@ -1078,8 +1082,9 @@
\end_layout

\begin_layout Standard
+\align right
\begin_inset Formula $$-u_{i}=\frac{\kappa}{R_{0}}u_{i}^{'}\label{eq:u dim}$$
+\rho=\rho_{0}\rho^{'}\label{eq:rho dim}

\end_inset

@@ -1089,7 +1094,7 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-T_{0}=\Delta TT_{0}'\label{eq:T0 dim}$$
+\alpha=\alpha_{0}\alpha^{'}\label{eq:alpha dim}

\end_inset

@@ -1099,7 +1104,7 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-T=\Delta T(T'+T_{0}')\label{eq:T dim}$$
+g=g_{0}g^{'}\label{eq:g dim}

\end_inset

@@ -1109,17 +1114,23 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-t=\frac{R_{0}^{2}}{\kappa}t^{'}\label{eq:t dim}$$
+\kappa=\kappa_{0}\kappa^{'}\label{eq:kappa dim}

\end_inset

+\begin_inset Formula $$+\eta=\eta_{0}\eta^{'}\label{eq:eta dim}$$
+
+\end_inset
+
+
\end_layout

\begin_layout Standard
\align right
\begin_inset Formula $$-H=c_{P0}\Delta TH^{'}\label{eq:H dim}$$
+c_{P}=c_{P0}c_{P}^{'}\label{eq:cp dim}

\end_inset

@@ -1129,7 +1140,7 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-\eta=\eta_{0}\eta^{'}\label{eq:eta dim}$$
+x_{i}=R_{0}x_{i}^{'}\label{eq:x dim}

\end_inset

@@ -1137,9 +1148,18 @@
\end_layout

\begin_layout Standard
+\begin_inset Formula $$+u_{i}=\frac{\kappa_{0}}{R_{0}}u_{i}^{'}\label{eq:u dim}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
\align right
\begin_inset Formula $$-P=\frac{\eta_{0}\kappa}{R_{0}^{2}}P^{'}\label{eq:p dim}$$
+T_{0}=\Delta TT_{0}'\label{eq:T0 dim}

\end_inset

@@ -1147,16 +1167,64 @@
\end_layout

\begin_layout Standard
+\align right
+\begin_inset Formula $$+T=\Delta T(T'+T_{0}')\label{eq:T dim}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\align right
+\begin_inset Formula $$+t=\frac{R_{0}^{2}}{\kappa_{0}}t^{'}\label{eq:t dim}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\align right
+\begin_inset Formula $$+H=c_{P0}\Delta TH^{'}\label{eq:H dim}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\align right
+\begin_inset Formula $$+P=\frac{\eta_{0}\kappa_{0}}{R_{0}^{2}}P^{'}\label{eq:p dim}$$
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
\noindent
where
\begin_inset Formula $\rho_{0}$
\end_inset

is the reference density,
-\begin_inset Formula $R_{0}$
+\begin_inset Formula $\alpha_{0}$
\end_inset

- is the radius of the Earth,
+ is the reference thermal expansivity,
+\begin_inset Formula $g_{0}$
+\end_inset
+
+ is the reference gravity,
+\begin_inset Formula $\kappa_{0}$
+\end_inset
+
+ is the reference thermal diffusivity,
\begin_inset Formula $\eta_{0}$
\end_inset

@@ -1164,18 +1232,21 @@
\begin_inset Formula $c_{P0}$
\end_inset

- is the reference heat capacity, and
+ is the reference heat capacity,
+\begin_inset Formula $R_{0}$
+\end_inset
+
+ is the radius of the Earth, and
\begin_inset Formula $\Delta T$
\end_inset

- is the superadiabatic temperature drop from the core-mantle boundary (CMB)
- to the surface.
+ is the temperature drop from the core-mantle boundary (CMB) to the surface.
Dropping the primes, the equations become:
\end_layout

\begin_layout Standard
\begin_inset Formula $$-u_{i,i}+\frac{d\ln\bar{\rho}}{dr}u_{r}=0\label{eq:non-dimensional continuity eqn}$$
+u_{i,i}+\frac{1}{\bar{\rho}}\frac{d\bar{\rho}}{dr}u_{r}=0\label{eq:non-dimensional continuity eqn}

\end_inset

@@ -1184,7 +1255,7 @@

\begin_layout Standard
\begin_inset Formula $$--P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+(Ra\bar{\rho}T+Rab\Gamma-RacC)\delta_{ir}=0\label{eq:non-dimensional momentum eqn}$$
+-P_{,i}+\left(\eta(u_{i,j}+u_{j,i}-\frac{2}{3}u_{k,k}\delta_{ij})\right)_{,i}+(Ra\bar{\rho}\alpha T+Rab\Gamma-RacC)g\delta_{ir}=0\label{eq:non-dimensional momentum eqn}

\end_inset

@@ -1194,7 +1265,7 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-T_{,t}+u_{i}T_{,i}=T_{,ii}+H+(TODO)\label{eq:non-dimensional energy eqn}$$
+\bar{\rho}c_{P}\left(T_{,t}+u_{i}T_{,i}\right)\left(1+2\Gamma\left(1-\Gamma\right)\frac{\gamma_{ph}^{2}}{d_{ph}}\frac{Rab}{Ra}Di\left(T+T_{0}\right)\right)=\bar{\rho}c_{P}\kappa T_{,ii}+\rho\alpha gu_{r}Di\left(T+T_{0}\right)\left(1+2\Gamma\left(1-\Gamma\right)\frac{\gamma_{ph}}{d_{ph}}\frac{Rab}{Ra}\right)+\frac{Ra}{Di}\Phi+\bar{\rho}H\label{eq:non-dimensional energy eqn}

\end_inset

@@ -1207,13 +1278,13 @@
\emph on
Ra
\emph default
-, a Rayleigh number, is defined as:
+, the thermal Rayleigh number, is defined as:
\end_layout

\begin_layout Standard
\align right
\begin_inset Formula $$-Ra=\frac{\rho_{0}g\alpha\Delta TR_{0}^{3}}{\eta_{0}\kappa}\label{eq:Ra, a Rayleigh number}$$
+Ra=\frac{\rho_{0}g_{0}\alpha_{0}\Delta TR_{0}^{3}}{\eta_{0}\kappa_{0}}\label{eq:Ra, Rayleigh number}

\end_inset

@@ -1223,7 +1294,7 @@
\begin_layout Standard
\noindent
This is not the usual definition of the Raleigh number that is based on
- layer thickness; it is
+ layer thickness; it is based on the radius of the Earth
\begin_inset Formula $R_{0}.$
\end_inset

@@ -1260,13 +1331,28 @@
\begin_layout Standard
\align right
\begin_inset Formula $$-Rac=Ra\frac{\delta\rho_{ch}}{\rho_{0}}\label{eq:Rac, a chemical Rayleigh number}$$
+Rac=Ra\frac{\delta\rho_{ch}}{\rho_{0}}\label{eq:Rac, chemical Rayleigh number}

\end_inset

\end_layout

+\begin_layout Standard
+(TODO: Sue, I want this sentence in the same paragraph as the previous sentecnes.
+ Can you help?)
+\shape italic
+Di
+\shape default
+ is the dissipation number and is defined as:
+\begin_inset Formula $$+Di=\frac{\alpha_{0}g_{0}R_{0}}{c_{P0}}\label{eq:Di, dissipation number}$$
+
+\end_inset
+
+
+\end_layout
+
\begin_layout Section
Numerical Methods
\end_layout